Instantly, adv.
1. Without the least delay or interval; at once; immediately. Macaulay.
2. With urgency or importunity; earnestly; pressingly. "They besought him instantly." Luke vii. 4.
Syn. -- Directly; immediately; at once. See Directly.
Shades of meaning
Fernald has no entry of its own for instantly; he lists it under immediately.
Where he writes about it:
Even immediately is sliding from its instantaneousness, so that we are fain to substitute at once, instantly, etc., when we would make promptness emphatic. — from immediately